Problem 94 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The domain of the function \(f(x)=\ln (1-2|\cos x|)\) \(+e^{\cos -1(2 x / \pi)}\) is (A) \(\left(-\frac{\pi}{2}, \frac{-\pi}{3}\right) \cup\left(\frac{\pi}{3}, \frac{\pi}{2}\right)\) (B) \(\left[-\frac{\pi}{2}, \frac{-\pi}{3}\right) \cup\left[\frac{\pi}{3}, \frac{\pi}{2}\right]\) (C) \(\left[-\frac{\pi}{2},-\frac{\pi}{3}\right) \cup\left(\frac{\pi}{3}, \frac{\pi}{2}\right]\) (D) None of these

Step-by-Step Solution

Answer

The domain is \( \left( -\frac{\pi}{2}, -\frac{\pi}{3} \right) \cup \left( \frac{\pi}{3}, \frac{\pi}{2} \right) \), which does not exactly match any listed option.

Step 1: Analyze the function

The function is given as \( f(x) = \ln(1 - 2|\cos x|) + e^{\cos^{-1}(2x/\pi)} \). It consists of two parts: the natural logarithm \( \ln(1 - 2|\cos x|) \) and the exponential function \( e^{\cos^{-1}(2x/\pi)} \). We need to consider the domain restrictions for both parts.

Step 2: Determine the domain for \( \ln(1 - 2|\cos x|) \)

For the natural logarithm part \( \ln(1 - 2|\cos x|) \), the expression inside the logarithm must be positive: \( 1 - 2|\cos x| > 0 \). This simplifies to \( |\cos x| < 0.5 \). This inequality implies \( -0.5 < \cos x < 0.5 \). We can solve these inequalities by considering the cosine function's range.

Step 3: Solve \( -0.5 < \cos x < 0.5 \)

The values of \( x \) where \( -0.5 < \cos x < 0.5 \) corresponds to the intervals \( \left(\frac{\pi}{3}, \frac{2\pi}{3}\right) \cup \left(\frac{4\pi}{3}, \frac{5\pi}{3}\right) + 2n\pi \) for any integer \( n \). We focus on one period \( [0, 2\pi] \) and find within this range the valid intervals.

Step 4: Determine the domain for \( e^{\cos^{-1}(2x/\pi)} \)

The \( \cos^{-1} \) function is defined for \(-1 \leq 2x/\pi \leq 1\). Solving for \( x \), we obtain \(-\pi/2 \leq x \leq \pi/2\). This provides a range for \( x \) that we will intersect with the interval found in Step 3.

Step 5: Find the intersection of intervals

Combine the valid intervals from \( \ln(1 - 2|\cos x|) \), \( \left(\frac{\pi}{3}, \frac{2\pi}{3}\right) \), and from \( e^{\cos^{-1}(2x/\pi)} \), \( [-\pi/2, \pi/2] \). The intersection of these domains is \( \left( \frac{\pi}{3}, \frac{\pi}{2} \right) \cup \left( -\pi/2, -\frac{\pi}{3} \right) \), corresponding closely with one of the given options.

Key Concepts

Natural logarithm

The natural logarithm, denoted as \( \ln(x) \), is a mathematical function that appears frequently when discussing domains in calculus and complex numbers. Essentially, the natural logarithm is the power to which the base, Euler's number \( e \), must be raised to produce a given number. For \( \ln(1-2|\cos x|) \) to be valid, the expression \( 1-2|\cos x| \) must be greater than zero because the logarithm of a negative number or zero is not defined in the set of real numbers.
This condition ensures the range of \( \ln(x) \) remains within real numbers and is not imaginary. In the context of our problem, it is necessary to find the intervals where \( |\cos x| < 0.5 \). Solving the inequality \( 1-2|\cos x| > 0 \) will give us the valid set of \( x \) values, revealing the domain where the natural logarithm part of the function lives. This step is crucial for defining the overall domain of the function.

Inverse trigonometric functions

Inverse trigonometric functions are used to derive the angle which corresponds to a given trigonometric ratio. Using the cosine inverse function, denoted as \( \cos^{-1}(x) \), we determine the angle whose cosine is \( x \). The restricted domain of the cosine inverse function is \( -1 \leq x \leq 1 \), meaning the arguments passed to \( \cos^{-1} \) must lie within this range to maintain its definition within the set of real numbers.
For the function \( e^{\cos^{-1}(2x/\pi)} \), it's essential that the argument \( \frac{2x}{\pi} \) stays within these bounds, which implies solving the inequality \( -1 \leq \frac{2x}{\pi} \leq 1 \) to find permissible values of \( x \).

• Solving \(-1 \leq \frac{2x}{\pi} \leq 1 \) will yield \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \).

By maintaining this range for \( x \), it ensures that the domain properly adheres to the constraints imposed by inverse trigonometric functions.

Exponential function

The exponential function, usually denoted as \( e^x \), is a fundamental mathematical concept where \( e \) (approximately 2.718) is raised to the power of \( x \). It's an intriguing function because its rate of growth increases exponentially, meaning it can grow quite rapidly. The exponential function is always defined for all real numbers, which makes it very versatile. However, when combined with other functions, such as \( \cos^{-1}(x) \) in our exercise, we need to carefully consider the domain of the combined functions.
In \( e^{\cos^{-1}(2x/\pi)} \), the main role of the exponential is to keep the result positive, as \( e^{x} \) always outputs values greater than zero, regardless of the input \( x \). This ensures that no matter what angle is returned from \( \cos^{-1} \), the exponential component will boost it into the positive domain.

• This characteristic helps define possible values of the overall function, taking into account previously established boundaries from \( \ln \) and inverse trigonometric calculations.

Given all these, the exponential portion's domain in isolation is not restricted, but its interplay with other elements determines the ultimate valid ranges for the full function.